Free product decompositions in images of certain free products of groups
by
Nikolay Romanovskiy
Institute of Mathematics, Novosibirsk, Russia
Coauthors: John S. Wilson (University of Oxford)

In 1978 the speaker proved the following result:

Let G be a group which has a presentation with n generators x₁, ..., x_n and m relators, where m < n, and let S = {x₁, ..., x_n}. Then some subset of n-m elements of S freely generates a free group.

The history of this result dates back to 1930, when Magnus published his Freiheitssatz, which is essentially the case of our statement in which m=1. In 2004 J.S.Wilson generalized above-mentioned result by proving a similar statement in which S is any generating set for G. The proof was indirect, relying on another result of the speaker.

Here we give a direct proof of a considerably more general result. Roughly speaking, the improvement consists of the replacement of the elements x_i by subgroups, of the members of S by suitably small subgroups, and of the hypothesis that S generates G by a weaker hypothesis.

Date received: January 20, 2006